Problem 4. BFS and DFS Trees. Let G be a connected graph with the property that both BFS and DFS starting from some node v return the same tree. Complete the proof below to argue that G itself must be a tree. In the proof, use only the facts about non-tree edges in BFS and DFS trees that we proved in class: i.e., fact (3.4) on page 81 and fact (3.7) on page 85 of Kleinberg and Tardos..

Problem 4. BFS and DFS Trees. Let G be a connected graph with the property that both BFS and DFS starting from some node v return the same tree. Complete the proof below to argue that G itself must be a tree. In the proof, use only the facts about non-tree edges in BFS and DFS trees that we proved in class: i.e., fact (3.4) on page 81 and fact (3.7) on page 85 of Kleinberg and Tardos..

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Recall that a subgraph H of a graph G is proper if H + G. a) Prove that if a connected graph G is a tree, then no proper spanning subgraph of G is connected. b) [NOTE: Can be solved without having done part a)!] Now prove the converse: if G is a connected graph such that no proper spanning subgraph of G is connected, then G is a tree.
Give an example of a connected, weighted, undirected graph such that neitherthe BFS-tree nor the DFS-tree is an MST, regardless of how the adjacency lists are orderedand at which vertex BFS or DFS are started. Argue that your graph satisfies the aboverequirements.
For the sake of argument, let's say we do a DFS on a directed graph Gd, where G is the depth-first tree or forest of Gd. If we get rid of all the back edges with respect to Gd, the resulting graph G will have no cycles. Your thoughts?
Let G be a connected graph with n vertices and m edges.Which of the following statements are true?(i) G is a tree if and only if n = m+1.(ii) G is a tree if and only if m = n+1.(iii) G is a tree if and only if the addition of any edgeto G will produce a unique cycle.(iv) G is a tree if and only if G contains at least onepath b etween any two vertices.(v) If G is connected, then G is a tree if and only if Gcontains at most one path between any two vertices.(vi) A connected subgraph of a tree is always a tree.
Given a DAG and two vertices v and w, find the lowest commonancestor (LCA) of v and w. The LCA of v and w is an ancestor of v and w that has nodescendants that are also ancestors of v and w. Computing the LCA is useful in multipleinheritance in programming languages, analysis of genealogical data (find degree ofinbreeding in a pedigree graph), and other applications. Hint : Define the height of avertex v in a DAG to be the length of the longest path from a root to v. Among verticesthat are ancestors of both v and w, the one with the greatest height is an LCA of v and w.
A complete graph G with 5 nodes has positive weight edges, each node has a distinct weight with an integer value and maximum weight is equal to number of edges in G. What can be the maximum weight of minimum spanning tree for graph G?
H.w: show that Let x be a e dge of Connected graph G. The following statements are equivalent. (1) X is X is a bridge of G (2) X is not on any eycle of G ® There exist vertices u and vof G such the ecdgex is on every path goning u and V O There exists a pardition of V into su bsets ų and w such that for any verticesUEU and w the edge x ison every path joining u and w
Suppose we do a BFS on a connected undirected graph G and T is corresponding depth-first tree. if we remove from G all the cross edges with respecr to T the result will just be a tree. True or False?
Recall that the degree d(u) of a node u in a graph is the number of neighbors of u. Prove the following statements. (a) In any graph, the number of nodes with odd degree must be even. (b) Every tree contains at least two nodes with degree 1. (c) If every node of a graph G has degree at least n/2, then G is connected.
Let G = (V, E) be a connected, undirected graph, and let s be a fixed vertex in G. Let TB be the spanning tree of G found by a bread first search starting from s, and similarly TD the spanning tree found by depth first search, also starting at s. (As in problem 1, these trees are just sets of edges; the order in which they were traversed by the respective algorithms is irrelevant.) Using facts, prove that TB = TD if and only if TB = TD = E, i.e., G is itself a tree.
Consider a connected graph G with at least 4 edges that has all distinct edge weights. Which of the following properties must be true of a Minimum Spanning Tree (MST) of G? I. The MST must contain the shortest edge of G. II. The MST must contain the second-shortest edge of G. III. The MST can never contain the longest edge of G. None I Only I and II Only I and III Only O1, II, and III
a) Draw the connected subgraph of the given graph above which contains only four nodes ACGB and is also a minimum spanning tree with these four nodes. What is its weighted sum? Draw the adjacency matrix representation of this subgraph (use boolean matrix with only 0 or 1, to show its adjacency in this case).b) Find the shortest path ONLY from source node D to destination node G of the given graph above, using Dijkstra’s algorithm. Show your steps with a table as in our course material, clearly indicating the node being selected for processing in each step.c) Draw ONLY the shortest path obtained above, and indicate the weight in each edge in the diagram. Also determine the weighted sum